Application of Semi-Integral Electroanalysis to the Study of the Kinetics of the Electrode Process Robert S. Rodgers Department of Chemistry, Lehigh University, Bethlehem, Pa. 780 75
It is suggested that electrochemical rate parameters may be easily determined by simultaneous measurement of current, i, and its semi-integral, m, in a potential-step experiment. A plot of i vs. m is a straight line whose slope and intercept are simply related to the forward and reverse electrochemical rate constants. A square current pulse may be used to cancel the effects of the double-layer charging current. Analog semi-integration would allow Immediate display of a plot of i vs. m, and determination of the rate constants without sophisticated computation.
ing rate data taken over times such that both diffusion and the kinetics of the electrode reaction are equally important.
THEORY The semi-integral of the faradaic current is simply related to the concentrations of the electroactive species a t the surface of the electrode. Consider the quasi-reversible reaction
Ox
kf f
iic-
Red
kb
For a typical potential-step experiment, the potential of Recently, some attention has been paid to the use of the "semi-integral" of current in electrochemical measurements (1-6). Much of this attention has been focused on its application to cyclic voltammetry in a n attempt to make quantitative data more easily obtainable. The purpose of this communication is t o point out a benefit of using the semi-integral in the study of electrode kinetics. Potential-step chronoamperometry and chronocoulome t 9 have often been used to elucidate electrode mechanisms and to measure the associated rate constants (7, 8). One problem associated with these techniques is the difficulty of data analysis. T o utilize the data taken during the time when mass transport and the electrode kinetics are both important factors in controlling the current (or the rate of charge accumulation), non-linear regression analysis has been applied to the i-t and q-t curves ( 7 , 9 ) .
i = K exp(X2t) e r f c ( ~ t ' / * ) q = (K/X2)[exp(X2t)erfc ( ~ t ' " ) + 2 ~ t ' / * / n ~ / *
the electrode is stepped from some relatively positive, equilibrium potential to one where the reduction is initially kinetically controlled. If the solution was homogeneous prior to the potential step, then after the step ( 2 ) .
where m = m ( t ) is the time-dependent semi-integral of the faradaic current, and the superscript "O" refers to surface concentrations, while the superscript "h" refers to the initial, bulk concentration. Other symbols have their usual meaning. Note that because of the nature of the semi-integral, these relations are valid, independent of the path taken from the initial potential to the potential of interest (2).
If the reduction is irreversible, then the current is related to the surface concentrations of Ox and Red (Coox and
c'Ked).
i = I z F A ~ , C " , - i7F24kbCoRed
(4)
and k f = kos,h exp(-a-[E nF RT k, =
exp([1
-
-
1.2 F a ] -RT [E
E"] -
E'])
Because of the complexity of these functions, use of a computer (either on line or off) is a necessity. Some decrease in the complexity can be gathered by only using chronocoulometric data obtained a t long times, once the current is diffusion controlled ( 7 ) .
2Kt'I2
K
q e p T y - 2
(if A t > 2 . 5 ) and performing linear regression analysis to the q-t 1/2 data. This has the disadvantage of ignoring the data taken a t the time when the factor of interest, the kinetics of the electrode reaction, is most important. Analog implementation of the semi-integral of the current ( 1 0 ) can afford a simple and rapid method of analyz-
i = ZI' - I J l X (5) where K and h have the same definitions as in Equations 3, a and b. The faradaic current and its semi-integral are simply related. Note t h a t this relation is independent of the nature of the experiment: The current is dependent upon the instantaneous values of K and X (and thus upon the instantaneous values of E and 42, the potential a t the Outer Helmholz Plane) and the value of the semi-integral, m. In the particular case of a potential-step experiment, a short time after the step-once the double-layer has been fully charged-K and X will be time independent and a plot of i us. m will be a straight line with an intercept of K and a slope of -h. This straight-line relation holds both in the region where the current is primarily kinetically controlled as well as the region where it is primarily diffusion controlled, Le., there is no restriction placed on the product At 112.
ANALYTICAL CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
281
Table I. Kinetic P a r a m e t e r s for the Reaction eEu(1II) 2 Eu(I1) in O.lFNaNCS, 0.01FLaC13. Hanging Hg Drop Electrode, Area = 0.035 cm2
+
E,
E9 Parameter
lib
mA-cni-'
x sec-1/2 l?l*b
paniplonibs -cni-'
m/A , w a m p l o m b - c m - 2
Faradaic current density vs. its semi-integral for ImF Eu(C104)3in 0.1 F NaNCS, 0.01 F LaCI3. Figure 1.
For clarity, not all data points are shown. El, +, E = -0.69 V ; 0 , E = 0.71 V
E = -0.65
V; A , E = -0.67 V;
Since analog semi-integration of the current can be performed by a simple operational amplifier circuit ( I O ) , it is possible to display this plot of i LIS. m directly on a x-y oscilloscope or recorder. T h e rate parameters can be easily measured without the necessity of employing a computer to perform a complicated data analysis. Alternatively, if a computer is used, the rate parameters can be assessed in the region where the current is primarily kinetically controlled, as well as the region where diffusion is dominant, without the complexity of a non-linear regression analysis to Equation 2.
Used'
5 2 5 2 5 2
-0.65
1.59 1.52 11.5 10.8 138 141
Std
v
emor of
-0.67
-0.69
-0.71
parameter
2.17 2.05 13.2 12.3 164 167
3.00 2.79 16.4 15.3 180 182
4.51 3.98 23.8 20.9 189 192
0.02 0.02 0.2 0.1 1 2
0 Equation 5, data taken every 2 msec, analysis performed on data taken between 2 and 38 msec (19 data points). Equation 2, data taken every 0.4 msec, analysis performed on data taken between 2 and 40 msec (96 data points). Both K and m* have been normalized to unit electrode area and represent current and semiintegral densities. k f = K/nFCbox. 1 amplomb = 1 coulombsec-1 * = 1 ampere-secl *.
\,
R E S U L T S A N D DISCUSSION Both the current and its semi-integral were derived from digitized charge-time data obtained for a solution of 1 m F Eu(C104)J in 0.1F NaNCS, 0.01F LaC13. Experimental procedure can be found in Reference 8 A computerized digital data acquisition system ( I I ) was used to control the experiment and acquire the charge-time data. All potentials are referenced to a conventional SCE reference electrode which had been filled with saturated XaC1 instead of
KCI. Digitized values of charge were obtained a t evenly spaced time intervals (2 msec). Consecutive triplets of these values of Q ( t = I A t ) ; I = 0 , 1, 2. . . .; were fit to a second degree polynomial.
+
4(1) = a,t'
h,t
+
cI
(6)
+
for ( I - 1) At < t < i l 1)At where I = 1, 2 , . . . . Differentiation of Equation 6 gave the current over the same interval
i ( n = 2a,l
+
h,
(I
-
1 ) ~< i
t
